Spectra of edge-independent random graphs

Abstract

Let G be a random graph on the vertex set {1,2,...,n} such that edges in G are determined by independent random indicator variables, while the probabilities pij for {i,j} being an edge in G are not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of G are recently studied by Oliveira and Chung-Radcliffe. Let A be the adjacency matrix of G, Ā=E(A), and Δ be the maximum expected degree of G. Oliveira first proved that asymptotically almost surely ||A-Ā||=O(√Δlnn) provided Δ≥Clnn for some constant C. Chung-Radcliffe improved the hidden constant in the error term using a new Chernoff-type inequality for random matrices. Here we prove that asymptotically almost surely ||A-Ā||≤(2+o(1))√Δ with a slightly stronger condition Δ≫ln4n. For the Laplacian matrix L of G, Oliveira and Chung-Radcliffe proved similar results ||L-L-||=O(√lnn/√δ) provided the minimum expected degree δ≥C′lnn for some constant C′; we also improve their results by removing the √lnn multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classical Erdo{double acute}s-Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs.